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Monotone Completeness Conjecture

The monotone completeness conjecture asserts that every AW*-algebra is monotone complete, meaning that every increasing norm-bounded net of self-adjoint elements has a supremum in its self-adjoint part (Blackadar 2006). Monotone sequential completeness imposes this requirement only on sequences.

Arulseelan and Hanson (2026) refute the conjecture by constructing a finite AW*-algebra with an increasing bounded sequence of positive elements having no supremum. The example is a type II_1 AW^*-factor and is obtained from a modified construction related to Kaplansky's quasitrace conjecture. The properly infinite case remains open. Here properly infinite means that the unital algebra contains two isometries with orthogonal ranges.

See also

AW*-Algebra, Kaplansky's Quasitrace Conjecture, Supremum, von Neumann Algebra

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References

Arulseelan, J. and Hanson, J. E. "Counterexamples to the Monotone Completeness Conjecture." 19 Oct 2026. https://arxiv.org/abs/2610.27184.Blackadar, B. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Berlin, Germany: Springer, 2006. https://doi.org/10.1007/3-540-28517-2.

Cite this as:

Weisstein, Eric W. "Monotone Completeness Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MonotoneCompletenessConjecture.html

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